Abstract. Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying $V=\HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq\HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form $\Ehrenfeucht(A,P,Q)$, which asserts that whenever an object $b$ is definable in $M$ from some $a\in A$ using parameters in $P$, with $b\neq a$, then the types of $a$ and $b$ over $Q$ in $M$ are different. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set $b$ is \emph{algebraic} in $a$ if it is a member of a finite set definable from $a$ (as in J. D. Hamkins and C. Leahy, Algebraicity and implicit definability in set theory). Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis, we answer two open questions posed in my paper with Leahy, by showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

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Dear Joel,
this is just a couple of words related to this profound study.
Sorry for taking your time.
I don’t have in hands the ND paper, so whatever follows is related only to the 1501.01918 version.
1. Theorem 4.6 (perhaps by Groszek and/or Laver) was somewhat sharpened in Golshani-K-Lyubetsky, MLQ, 63, No. 1–2, 19–31 (2017): there is a CCC extension
$L[a,b]$ of $L$ by a pair of reals $a,b$, such that 1) the Vitali (or $E_0$) classes A of a
and B of b are different (countable) sets, 2) A and B are OD-indiscernible in $L[a,b]$,
and 3) $A\cup B$ is lightface $\Pi^1_2$ (the lowest possible).
2. Question 4.12 on p 12, its meaning is somewhat elusive. We assume that any OD-algebraic set is OD, and we want to know whether for any two OD sets $x\ne y$, if
$y$ is parameter-free algebraic wrt $x$ then $x$, $y$ are necessarily parameter-free discernible. Is this the idea or I am taking it wrong?
Best
Vladimir